- Moore’s law of cameras: LSST 3.2 gigapixel camera photos released
**HW:**Read Ch. 1-4

- Nature review on Numpy
**HW:**Read Ch. 5-7

- Root finding summary slides (based on Numerical Recipes book)
- Root finding convergence, Newton fractal notebook and zooming movie
- Project1 ideas
**HW:**Read Ch. 8-9.

- 2020 Nobel prize in physics
**Roger Penrose**invented ingenious mathematical methods to explore Albert Einstein’s general theory of relativity. He showed that the theory leads to the formation of black holes, those monsters in time and space that capture everything that enters them. Nothing, not even light, can escape.**Reinhard Genzel and Andrea Ghez**each lead a group of astronomers who have focused on a region at the centre of the Milky Way since the early 1990s. With increasing precision, they have mapped the orbits of the brightest stars that are closest to the centre. Both groups found something that is both invisible and heavy, forcing this jumble of stars to swirl around. This invisible mass has about four million solar masses squeezed together in a region no larger than our solar system. What is it that makes the stars at the heart of the Milky Way swing around at such astonishing speeds? According to the current theory of gravity, there is only one candidate – a supermassive black hole.

- If you are aspiring for Nobel prize you should take this class seriously! Estimation of the mass of the supermassive blackhole from observed positions of stars circling around it is a
**nice example**of data modelling, regression. :-) - A nice summary video (from 2009!) describing the discovery of our Milky Way’s supermassive black hole and the technologies that made it possible.
- Maximum Likelihood Estimation vs. Least Square Fit summary slides and notebook
**Can we improve simple algorithms like multiplication?**- https://www.quantamagazine.org/mathematicians-discover-the-perfect-way-to-multiply-20190411/
- https://hal.archives-ouvertes.fr/hal-02070778/document
- Strassen method: https://en.wikipedia.org/wiki/Strassen_algorithm
- Coppersmith–Winograd algorithm: https://en.wikipedia.org/wiki/Coppersmith%E2%80%93Winograd_algorithm)

- Chemistry Nobel Prize for CRISPR/Cas9 genome editing system
- Nemeth Robert’s question from last class:
*Where can I find a derivation for the fourth order Runge-Kutta method which follows a similar logic as the one for rk2 in Landau’s book (in 9.5.2.)?***Answer**: I thought that it is presented in Numerical Recipes, but it seems that derivation of the 4th order Runge-Kutta is more tedious than I thought. It is not derived in the NR book, but I have found this “simplified” derivation: https://www.researchgate.net/publication/49587610_A_Simplified_Derivation_and_Analysis_of_Fourth_Order_Runge_Kutta_Method

- Ordinary Differential Equations - explanation slides
- Predictor-Corrector Method explanation, local annotated copy
- Energy conservation notebook

**HW:**Read Ch. 10-11

- Room temperature superconductivity achieved for the first time - though not at room pressure
- 1986: first “high temperature” superconducting material

- Methods for
**differentiation and calculating gradients**, on emphasis on neural network weight optimization- gradient methods animated figure
- gradient methods described
- automatic differentiation, local annotated copy
- autodiff: https://justindomke.wordpress.com/2009/02/17/automatic-differentiation-the-most-criminally-underused-tool-in-the-potential-machine-learning-toolbox/
- automatic differentiation short explanation

- Visualization of multidimensional minimization Loss Landscape
- Machine learning as a useful tool for cosmology studies:
- Francisco Villaescusa-Navarro et al. : The CAMELS project: Cosmology and Astrophysics with MachinE Learning Simulations
- Machine learning to analyse gravitational lensing observations
- Machine learning to replace N-body simulations

- Machine learning in physics review: Carleo, G., Cirac, I., Cranmer, K., Daudet, L., Schuld, M., Tishby, N., Vogt-Maranto, L. and Zdeborová, L., 2019. Machine learning and the physical sciences. Reviews of Modern Physics, 91(4), p.045002. arxiv

- Fourier analysis
- Viusal interactive explanation of Fourier analysis
- IEEE Top 10 algorithms of the 20th century. Some details
- gravitational wave data analysis from LIGO Open Science Center
- “FFT” on a sphere: Cosmic Microwave Background radiation analysis. Detailed example notebooks
- Nyquist sampling theorem/ compressive sensing

- Project2 ideas
**HW:**Read Ch. 12

- Professional PDE codes:
- py-pde (relatively simple, python)
- Finite Volume: OpenFoam - Python wrapper
- Finite Element: FEniCS

- Project idea, “light saber” simulation
- simulate electromagnetci knots, that are closed field lines of both electric and magnetic field lines
- https://iopscience.iop.org/article/10.1088/2399-6528/aa9761
- https://journals.aps.org/pre/pdf/10.1103/PhysRevE.101.063305
- code 1: https://github.com/flaport/fdtd
- code 2: Meep

- Fermi-Ulam-Pasta-Tsingou system and original paper
- Mary Tsingou’s contribution to the Fermi-Ulam-Pasta study
- Logistic map (nice animations)
- In 1975, Dr. Feigenbaum, using the small HP-65 calculator discovered that the ratio of the difference between the values at which such successive period-doubling bifurcations occur tends to a constant of around 4.6692.
- Stretch-and-fold chaos and the horseshoe map
- Coupled chaotic maps
- Chaos in dripping faucet time series
**HW:**Read Ch. 13-14.

- Project1 evaluation summary
- Chaotic double pendulum notebook
- Lorenz attractor
- Dimensionality of the underlying dynamics: Taken’s theorem and the Grassberger-Procaccia algorithm
- make your own chaotic circuit: https://www.researchgate.net/publication/309351711_A_simple_chaotic_circuit_with_a_light-emitting_diode
- News: related to Grassberger-Procaccia algorithm and Takens theorem:
- ‘AI Copernicus ‘discovers’ that Earth orbits the Sun’ review
- paper
- Recovering Hamiltonian with machine learning link
- Asseman, A., Kornuta, T. and Ozcan, A., 2018. Learning beyond simulated physics.
- real experimental data to train machine learning

- Lotka-Volterra population dynamics model and original hare-lynx time series
- http://mc-stan.org/users/documentation/case-studies/lotka-volterra-predator-prey.html
- not only animal population
- COVID-19 “agent based modeling” vs. SIR model
**predator prey like systems in chemistry**:- Briggs–Rauscher oscillating reaction (nice video)
- Belousov–Zhabotinsky reaction (with simulation demo) and experiment video1 , video2
- Examples of more complex systems: Citric acid cycle , Metabolic pathway, “Google map” of metabolic pathways

**Chaotic simulation precision**- Sympletic integrators for nonlinear Hamiltonian systems: https://en.wikipedia.org/wiki/Symplectic_integrator
- Corless, R.M., 1994. What good are numerical simulations of chaotic dynamical systems?. Computers & Mathematics with Applications, 28(10-12), pp.107-121. link
- Li, X. and Liao, S., 2018. Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems. Applied Mathematics and Mechanics, 39(11), pp.1529-1546. link
- Boekholt, T.C.N., Portegies Zwart, S.F. and Valtonen, M., 2020. Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length. Monthly Notices of the Royal Astronomical Society, 493(3), pp.3932-3937. link :
*“… using the accurate and precise N-body code Brutus, which goes beyond standard double-precision arithmetic … three massive black holes with zero total angular momentum, we conclude that up to five percent of such triples would require an accuracy of smaller than the Planck length in order to produce a time-reversible solution”*

**HW:**Read Ch. 15

**NEWS:**The “holy grail” of molecular dynamics, protein folding is*solved*:**Fractals:**- Mandelbrot set
- News: Onsager prize, Tamas Vicsek
- fractal dimension
- box counting method
- fractal GPS path

- example fractal dimensions
- DNA fractal Peano-Hilbert globule article1 , article2
- Heart rate fluctuation dynamics fractal
- Arnold’s cat map
- fractal antenna
- genetic algorithm related to Barnsley’s fern and also evolutionary models
- Some fun with Diffusion Limited Aggregation
**Cellular automata:**- 1D
- NKS
- Wolfram Physics Project - Cellular Automata as building blocks of natural laws
- Golly
- Neumann universal constructor with a surprise
- Digits of Pi
- Gerard ‘t Hooft’s idea
- on general graph: Kauffman’s NK boolean network and some analysis
**High performance computing:**- Top 500, 2020 Fugaku, 500 petaFlops (5e17)
- In the
**NEWS:** - Bitcoin hashrate (2020) 150 exaHash/s, 1hash/s ~ 10kFlops, ~ 1.5 yottaFlops (1.5e24 Flops)
**HW:**Read Ch. 16,17

**NEWS:**- Quantum computing supremacy with boson sampling

**Thermodynamic simulations, Ising model:**- Simplest model of phase transition: Ising model, exact 2D solution, Onsager 1944 and a real life 2D ferromagnet (2018)
- Faster dynamics, spin clusters: Swendsen-Wang algorithm, visualization
- Scaling at the critical point, renormalization demonstration
- Metropolis algorithm and simulated annealing
- Hopfield model
- Neural nets have similar complex energy landscape: visualizations , article
- Numerical optimization and stat phys.: Moore & Mertens: The Nature of Computation , short summary
- NP-hardness concept, list of problems ; protein folding is NP-hard? or not

**NEWS 2020:**The “holy grail” of molecular dynamics, protein folding is*solved*:- Nature editorial
- DeepMind blog
- New aspects: Adiabatic quantum computation
- Simulated bifurcation
- the Feynman path integral is based on variational principle like Fermat’s optical principle

**Molecular dynamics**- long range force cutoff, Ewald: link link2
- initial transient. Start with Maxwell distr. but since the coordinate-velocity correlations are not correct, temperature rescaling needed: link
- protein folding link
- energy landscape: link
- badly folded proteins: prions
- protein folding with machine learning: alphaFold
- hoomd-blue open source molecular dynamics simulator program
- argon with Hoomd
- two/three component mixing with Hoomd
- interesting problem: tetrahedron packing theoretical limit and hoomd simulation

**Partial differential equations**- classification and boundary conditions and explanation of names
- Green’s Function explained with linear algebra analogy: @Artem at math.stackexchange
- Finite element method and comparision to Finite difference method link
- FTCS and Crank-Nicolson methods as stepping examples
- von Neumann stability analysis link
- openFOAM open source computational fluid dynamics program
- FEniCS open source finite element numerical solver for PDEs
- Meep open source finite-difference time-domain PDE solver
- Python 3D FDTD Simulator another open source finite-difference time-domain PDE solver (with PyTorch backend)

**HW:**Read the remaining chapters, prepare for the oral exam: short presentation of projects + questions related to the book’s topics